Telegraph equation

This article deals with the telegraph equation of electrodynamics . The (more specific) telegraph equation for the propagation of current and voltage on a line is dealt with under the line equation .

The telegraph equation is a general form of the wave equation . It is a 2nd order partial differential equation .

General

The telegraph equation is a partial differential equation (with hyperbolic, with elliptical and with parabolic) and reads in the general form ${\displaystyle a>0}$ ${\displaystyle a<0}$ ${\displaystyle a=0}$

${\displaystyle \Delta {\vec {F}}=a\cdot {\frac {\partial ^{2}{\vec {F}}}{\partial t^{2}}}+b\cdot {\frac {\partial {\vec {F}}}{\partial t}}+c\cdot {\frac {\partial {\vec {F}}}{\partial x}}+d\cdot {\vec {F}}}$.

Here is the Laplace operator , i.e. in a spatial dimension . The derivation according to here is representative of the derivation according to location coordinates. A scalar can also be used instead of a vector . ${\displaystyle \Delta {\vec {F}}}$ ${\displaystyle \Delta {\vec {F}}={\frac {\partial ^{2}{\vec {F}}}{\partial x^{2}}}}$ ${\displaystyle x}$ ${\displaystyle F}$

In this form it is an equation that contains many other linear partial differential equations in physics as special cases ( wave equation , diffusion equation , Helmholtz equation , potential equation ).

Telegraph equation with a> 0, b> 0; c = d = 0

The equations are generally of the type:

${\displaystyle \Delta {\vec {F}}=a{\frac {\partial ^{2}{\vec {F}}}{\partial t^{2}}}+b{\frac {\partial {\vec {F}}}{\partial t}}}$

The prefactor has the dimension of an inverse speed square. ${\displaystyle a}$

For example, the material equations of electrodynamics can be used to describe the Maxwell equations in charge-free space regions as

${\displaystyle \Delta {\vec {E}}={\frac {\mu \varepsilon }{c^{2}}}{\frac {\partial ^{2}{\vec {E}}}{\partial t^{2}}}+\sigma \mu _{0}\mu {\frac {\partial {\vec {E}}}{\partial t}}}$

and

${\displaystyle \Delta {\vec {H}}={\frac {\mu \varepsilon }{c^{2}}}{\frac {\partial ^{2}{\vec {H}}}{\partial t^{2}}}+\sigma \mu _{0}\mu {\frac {\partial {\vec {H}}}{\partial t}}}$.

where (c the speed of light in vacuum) was used. ${\displaystyle c^{2}={\frac {1}{\mu _{0}\,\varepsilon _{0}}}}$

These are wave equations for a lossy dielectric. In the case of an isolator and the Maxwell equations reduce to the (vectorial) wave equation. ${\displaystyle \sigma =0}$

Telegraph equation with a> 0; b = c = d = 0

The equations are generally of the wave equation type:

${\displaystyle \Delta F=a{\frac {\partial ^{2}F}{\partial t^{2}}}}$

In particular, one obtains the original by Oliver Heaviside introduced Telegraph equations for the voltage and the current in a double line with inductance and capacity (based on the length and generally spatially dependent): ${\displaystyle U}$ ${\displaystyle I}$ ${\displaystyle L}$ ${\displaystyle C}$

${\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}=L\,C{\frac {\partial ^{2}U}{\partial t^{2}}}}$

or.

${\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}=L\,C{\frac {\partial ^{2}I}{\partial t^{2}}}}$

where line losses were neglected. Since the wave propagates at the speed of. ${\displaystyle a=L\,C}$ ${\displaystyle {\frac {1}{\sqrt {(L\,C)}}}}$

Another example are the wave equations of the electromagnetic field given above in the case of no losses ( as in free space). ${\displaystyle \sigma =0}$

literature

• Adolf J. Schwab : Conceptual world of field theory. Practical, clear introduction. Electromagnetic fields, Maxwell's equations, gradient, rotation, divergence, finite elements, finite differences, equivalent charge method, boundary element method, moment method, Monte Carlo method. 6th unchanged edition. Springer-Verlag, Berlin et al. 2002, ISBN 3-540-42018-5 .

Web links

• Telegraph equation, Lexicon of Physics, Spektrum Verlag